Basics of Statistics and Mathematics Unit 5/8

Correlation analysis

A statistical technique used to measure the strength and direction of the relationship between two quantitative variables.

Coefficient of correlation (r)

A unit-less numerical value (−1 to +1) that quantifies the degree and direction of linear association between two variables.

Strength (of correlation)

Indicates how closely plotted data points cluster around a straight reference line; the nearer to ±1, the stronger the linear relationship.

Direction (of correlation)

Shows whether variables move together (positive) or in opposite directions (negative) when one variable changes.

Positive correlation

An association in which the values of two variables move in the same direction—both increase or both decrease.

Negative correlation

An association in which the values of two variables move in opposite directions—one increases while the other decreases.

Linear correlation

A relationship in which paired values change at a constant or proportional rate, forming a straight-line trend on a graph.

Non-linear (curvilinear) correlation

A relationship where paired values do not change proportionally; plotted points follow a curved pattern rather than a straight line.

Simple correlation

Study of the association between exactly two variables.

Partial correlation

Measurement of the association between two variables while holding the effects of additional influencing variables constant.

Multiple correlation

Study of the association involving more than two variables simultaneously.

Scatter diagram

A graph that plots paired values of two variables (x, y) to provide a visual impression of their relationship.

Covariance

A statistic measuring how two variables deviate together from their respective means; basis for Pearson’s r.

Karl Pearson’s correlation coefficient

A quantitative measure of linear association, computed as covariance divided by the product of the standard deviations of x and y.

Coefficient of determination (r²)

The proportion of variance in the dependent variable that is explained by the independent variable; ranges from 0 to 1.

Standard error of correlation (SEr)

An estimate of the sampling variability of r, given by SEr = √(1 − r²)/(√n).

Probable error of correlation (PEr)

A range estimate for the population correlation, calculated as 0.675 × SEr; used to judge the significance of r.

Spearman’s rank correlation coefficient (Rho)

A non-parametric measure of association for ordinal (ranked) data, defined as R = 1 − [6Σd² / n(n² − 1)].

Ordinal data

Categorical data placed in a meaningful order (rank) without equal or known intervals between categories.

Nominal data

Categorical data consisting of labels or names with no intrinsic order or quantitative value.

Auto-correlation coefficient

A statistic that measures correlation of a variable with itself across different time lags.

Lead time / Lag time

The time difference between a cause and its effect when analysing time-series relationships.

Hypothesis testing

A procedure that uses sample information to decide whether a statement about a population parameter should be accepted or rejected.

Null hypothesis (H₀)

The default assumption that no effect or no relationship exists; considered true until evidence suggests otherwise.

Alternate hypothesis (H₁)

The statement accepted if the null hypothesis is rejected; claims that a real effect or relationship exists.

Directional hypothesis

Specifies not only that a relationship exists but also its direction (e.g., positive, negative, greater than, less than).

Non-directional hypothesis

States that a relationship exists without specifying the direction of the effect.

Acceptance region

Range of test-statistic values for which the null hypothesis is not rejected.

Rejection (critical) region

Range of test-statistic values that lead to rejection of the null hypothesis.

Critical value

A tabled threshold that separates the acceptance region from the rejection region in hypothesis testing.

Parametric test

Statistical test that requires interval or ratio-scale data and assumes specific population distributions (e.g., t-test).

Non-parametric test

Statistical test that uses nominal or ordinal data and makes fewer distributional assumptions (e.g., chi-square, rank tests).

Standard error formula for r

SEr = √(1 − r²)/(√n); used to compute t = r√(n − 2)/√(1 − r²) for significance testing of correlation.

Regression Analysis

A statistical technique that establishes a functional, cause-and-effect relationship between one dependent variable and one or more independent variables.

Simple Regression

Regression involving exactly two variables—one dependent and one independent.

Multiple Regression

Regression involving one dependent variable and two or more independent variables.

Linear Regression

Regression in which the relationship between variables is represented by a straight-line equation; no variable is raised to a power higher than one.

Non-Linear Regression

Regression in which the relationship between variables follows a curved (e.g., parabolic) trend rather than a straight line.

Total Relationship

A regression model that includes all relevant variables affecting the dependent variable; typically multiple regression.

Partial Relationship

A regression model that considers only selected variables, excluding others deemed less relevant.

Regression Equation

An equation that expresses the average relationship between dependent and independent variables, e.g., Y = a + bX or X = a + bY.

Dependent Variable (Y)

The variable whose values are predicted or explained in regression analysis.

Independent Variable (X)

The variable(s) that influence or predict the dependent variable in regression analysis.

Regression Line

The line of best fit that represents the average relationship between dependent and independent variables on a scatter plot.

Principle of Least Squares

The rule that the regression line should minimize the sum of the squared differences (errors) between observed and predicted values.

Residual (Error)

The difference between an observed value and the value predicted by the regression equation.

Line of Best Fit

A line drawn through data points that minimizes the total squared residuals, representing the best average fit.

Regression Coefficient

A numerical value that measures the rate of change in the dependent variable for a one-unit change in the independent variable.

Coefficient byx

Symbol for the regression coefficient of Y on X; indicates change in Y for a one-unit change in X.

Coefficient bxy

Symbol for the regression coefficient of X on Y; indicates change in X for a one-unit change in Y.

Scatter Diagram

A graph of paired (X,Y) data points used to visualize the relationship prior to fitting a regression line.

Correlation Coefficient (r)

A statistic (-1 to +1) measuring the strength and direction of linear association between two variables.

Coefficient of Determination (r²)

The proportion of variance in the dependent variable explained by the independent variable(s).

Normal Equations

Algebraic equations derived from the least-squares principle to solve for the regression constants a and b.

Graphic Method

A regression method that draws a line of best fit visually on a scatter diagram; considered subjective.

Algebraic Method

A regression method that determines the exact regression equation mathematically via least-squares calculations.

Cause-and-Effect Relationship

A directional association where changes in an independent variable cause changes in a dependent variable.

Properties of Regression Coefficients

Rules such as both coefficients having the same sign, r = ±√(b1·b2), and independence from origin but not scale.

Angle Between Regression Lines (θ)

The angle whose tangent equals |(m1–m2)/(1+m1m2)|; reflects the degree of dependence between variables.

Two Regression Equations

Separate equations (Y on X and X on Y) needed because variables are not interchangeable as predictor and response.

Uses of Regression Analysis

Prediction, trend estimation, economic modeling (demand, supply, cost), and calculation of r and r².

Difference: Correlation vs. Regression

Correlation measures strength of association; regression models the relationship and allows prediction.

Spurious Correlation

An apparent correlation between variables caused by a third factor or random chance, not by direct relationship.

Intercept (a)

The value of the dependent variable when the independent variable equals zero; the point where the regression line crosses the Y-axis.

Slope (b)

The regression coefficient representing the expected change in the dependent variable per unit change in the independent variable.

Functional Relationship

A mathematical link between variables expressed by the regression equation.

Regression Toward the Mean

Galton’s original observation that extreme values tend to be followed by values closer to the average.

Least-Squares Estimates (ŷ, x̂)

Predicted values of Y or X calculated from their respective regression equations.

Best-Fitted Line

Another term for the regression line produced by the least-squares method.

Graphic vs. Algebraic Methods

Graphic method relies on visual fitting; algebraic method relies on mathematical computation for exact coefficients.

Regression Constant (a)

The intercept term in a regression equation; adjusts the line vertically to best fit the data.

Unit Change Interpretation

The concept that a regression coefficient shows the marginal (unit-to-unit) change in the dependent variable.

Scale Dependence

Property that regression coefficients change when units of measurement (scale) change, though they remain independent of origin.

Index Number

A statistical measure that shows changes in a variable or a group of related variables over time, location, or other characteristics.

Specified Average

An average calculated from a set of data for a particular purpose (e.g., price or quantity), used to build an index number.

Percentage Representation

The practice of expressing index numbers as percentages relative to a base value of 100 to highlight proportional change.

Base Period

The reference time period (usually set to index = 100) against which all other periods are compared in an index.

Current Period

The time period whose prices or quantities are compared with the base period when computing an index number.

Price Index

An index that measures changes in the general price level of a specified basket of goods and services over time.

Quantity Index

An index that measures changes in the physical volume (output or consumption) of goods over time.

Value Index

An index that measures changes in the total monetary value (Price × Quantity) of goods between two periods.

Wholesale Price Index (WPI)

An index that tracks changes in the prices of goods at the wholesale (producer) level.

Consumer Price Index (CPI)

Also called the cost-of-living index; measures average changes in prices paid by consumers for a basket of goods and services.

Purchasing Power of Money

The quantity of goods and services that a unit of currency can buy; often tracked via the CPI or GDP deflator.

Economic Barometer

A metaphor describing index numbers’ role in reflecting overall economic conditions such as inflation or growth.

Simple Aggregate Index

An unweighted index obtained by dividing the sum of current-period prices by the sum of base-period prices and multiplying by 100.

Price Relative

The ratio (P1/P0 × 100) that shows the percentage change in price of an individual item between two periods.

Simple Average of Price Relatives (Arithmetic Mean)

An index computed by averaging individual price relatives with the arithmetic mean.

Simple Average of Price Relatives (Geometric Mean)

An index computed by taking the antilog of the mean of log price relatives; preferred for proportional changes.

Weighted Aggregate Index

An index that applies explicit weights to items to reflect their relative importance before aggregation.

System of Weighting

The method of assigning importance (weights) to individual items in an index number calculation.

Laspeyres Price Index

A weighted price index that uses base-period quantities (q0) as weights: Σp1q0 / Σp0q0 × 100.

Paasche Price Index

A weighted price index that uses current-period quantities (q1) as weights: Σp1q1 / Σp0q1 × 100.

Fisher’s Ideal Index

The geometric mean of the Laspeyres and Paasche indices; considered ‘ideal’ because it meets several consistency tests.

Bowley’s Index

The arithmetic mean of Laspeyres and Paasche indices: ½[(Σp1q0 / Σp0q0) + (Σp1q1 / Σp0q1)] × 100.

Marshall-Edgeworth Index

A weighted price index that uses the sum of base and current quantities (q0 + q1) as weights.

Kelly’s Index

A weighted price index that uses the average quantity (q̄ = (q0 + q1)/2) as the common weight.

Unit Test

A consistency check that an index formula should give the same result regardless of the units in which data are measured.

Time Reversal Test

A test requiring that an index computed forward and then backward in time yields a product of one: P01 × P10 = 1.

Factor Reversal Test

A test requiring that the product of a price index and a quantity index equals the corresponding value ratio.